Properties

Label 193200ef
Number of curves $6$
Conductor $193200$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("193200.df1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 193200ef

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
193200.df5 193200ef1 [0, -1, 0, 50392, -8934288] [2] 1572864 \(\Gamma_0(N)\)-optimal
193200.df4 193200ef2 [0, -1, 0, -461608, -103142288] [2, 2] 3145728  
193200.df3 193200ef3 [0, -1, 0, -2029608, 1013273712] [2, 2] 6291456  
193200.df2 193200ef4 [0, -1, 0, -7085608, -7257062288] [2] 6291456  
193200.df1 193200ef5 [0, -1, 0, -31653608, 68555993712] [2] 12582912  
193200.df6 193200ef6 [0, -1, 0, 2506392, 4896089712] [2] 12582912  

Rank

sage: E.rank()
 

The elliptic curves in class 193200ef have rank \(1\).

Modular form 193200.2.a.df

sage: E.q_eigenform(10)
 
\( q - q^{3} + q^{7} + q^{9} + 4q^{11} + 2q^{13} + 6q^{17} - 4q^{19} + O(q^{20}) \)

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.